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1 Distribution of Durations and Intervals

The distribution of element durations and inter-element intervals from the whale vocal sequences included in this analysis. The times are z-scored within each study to enable direct comparison.

Figure 1: The distribution of element durations and inter-element intervals from the whale vocal sequences included in this analysis. The times are z-scored within each study to enable direct comparison.

2 Effects in Whale Data

Table 1: The effect of sequence length on element/interval duration for each whale species, computed from the base model that excludes position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
Group	Species	Effect	2.5%	97.5%
Mysticete	Blue Whale	-0.255	-0.331	-0.178
	Bowhead Whale	-0.184	-0.318	-0.051
	Humpback Whale	-0.678	-0.692	-0.665
	Minke Whale	-0.278	-0.294	-0.261
	Right Whale	0.309	0.278	0.34
	Sei Whale	-0.213	-0.363	-0.063
Odontocete	Bottlenose Dolphin	-0.242	-0.347	-0.138
	Commerson's Dolphin	0.221	0.087	0.356
	Heaviside's Dolphin	-0.119	-0.32	0.083
	Hector's Dolphin	-0.008	-0.274	0.258
	Killer Whale	0.121	0.003	0.239
	Narrow-Ridged Finless Porpoise	-0.304	-0.338	-0.27
	Peale's Dolphin	-0.333	-0.489	-0.177
	Risso's Dolphin	-0.42	-0.448	-0.392
	Sperm Whale	-0.234	-0.241	-0.226

Table 2: The effect of sequence length and element/interval position on element/interval duration for each whale species, computed from the expanded model that includes both length and position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
		Length			Position
Group	Species	Effect	2.5%	97.5%	Effect	2.5%	97.5%
Mysticete	Blue Whale	-0.255	-0.331	-0.178	-0.064	-0.087	-0.041
	Bowhead Whale	-0.179	-0.313	-0.046	-0.751	-0.789	-0.713
	Humpback Whale	-0.678	-0.691	-0.665	-0.193	-0.201	-0.186
	Minke Whale	-0.278	-0.294	-0.261	-0.017	-0.023	-0.011
	Right Whale	0.309	0.278	0.34	0.107	0.096	0.119
	Sei Whale	-0.213	-0.364	-0.062	-0.105	-0.139	-0.071
Odontocete	Bottlenose Dolphin	-0.242	-0.346	-0.138	-0.084	-0.111	-0.056
	Commerson's Dolphin	0.221	0.087	0.356	-0.106	-0.118	-0.095
	Heaviside's Dolphin	-0.119	-0.32	0.083	0.019	0.01	0.027
	Hector's Dolphin	-0.008	-0.274	0.258	-0.001	-0.01	0.008
	Killer Whale	0.121	0.021	0.221	0.528	0.428	0.628
	Narrow-Ridged Finless Porpoise	-0.305	-0.339	-0.271	0.168	0.151	0.185
	Peale's Dolphin	-0.333	-0.489	-0.177	-0.013	-0.017	-0.009
	Risso's Dolphin	-0.42	-0.448	-0.392	-0.196	-0.2	-0.192
	Sperm Whale	-0.234	-0.241	-0.226	0.028	0.026	0.031

3 Effects in Human Data

Table 3: The effect of sequence length on element/interval duration for each human language, computed from the base model that excludes position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
Language	Effect	2.5%	97.5%
Anal	-0.104	-0.113	-0.095
Arapaho	0.03	0.02	0.04
Asimjeeg Datooga	-0.063	-0.073	-0.053
Baïnounk Gubëeher	-0.102	-0.11	-0.093
Beja	-0.066	-0.073	-0.059
Bora	-0.127	-0.138	-0.116
Cabécar	-0.11	-0.12	-0.1
Cashinahua	-0.1	-0.108	-0.091
Daakie	-0.131	-0.141	-0.121
Dalabon	-0.079	-0.091	-0.066
Dolgan	-0.13	-0.139	-0.121
English (Southern England)	-0.053	-0.066	-0.04
Evenki	-0.101	-0.109	-0.092
Fanbyak	-0.091	-0.101	-0.08
French (Swiss)	-0.05	-0.063	-0.037
Goemai	-0.124	-0.138	-0.11
Gorwaa	-0.125	-0.136	-0.114
Hoocąk	-0.099	-0.109	-0.09
Jahai	-0.062	-0.073	-0.05
Jejuan	-0.093	-0.104	-0.083
Kakabe	-0.123	-0.135	-0.111
Kamas	-0.096	-0.111	-0.082
Komnzo	-0.081	-0.091	-0.071
Light Warlpiri	-0.144	-0.154	-0.133
Lower Sorbian	-0.078	-0.089	-0.067
Mojeño Trinitario	-0.147	-0.156	-0.137
Movima	-0.014	-0.023	-0.006
Nafsan (South Efate)	-0.072	-0.082	-0.062
Nisvai	-0.064	-0.074	-0.053
Nllng	-0.098	-0.111	-0.086
Northern Alta	-0.069	-0.079	-0.059
Northern Kurdish (Kurmanji)	-0.022	-0.033	-0.011
Pnar	-0.021	-0.035	-0.007
Resígaro	-0.079	-0.089	-0.069
Ruuli	-0.039	-0.048	-0.03
Sadu	-0.124	-0.136	-0.112
Sanzhi Dargwa	-0.135	-0.148	-0.122
Savosavo	-0.052	-0.062	-0.042
Sümi	-0.09	-0.101	-0.079
Svan	-0.066	-0.074	-0.057
Tabaq (Karko)	-0.213	-0.222	-0.203
Tabasaran	-0.138	-0.151	-0.125
Teop	-0.17	-0.181	-0.159
Texistepec Popoluca	-0.07	-0.081	-0.059
Urum	-0.126	-0.134	-0.118
Vera'a	-0.152	-0.163	-0.141
Warlpiri	-0.147	-0.157	-0.137
Yali (Apahapsili)	-0.191	-0.202	-0.179
Yongning Na	-0.128	-0.142	-0.114
Yucatec Maya	-0.067	-0.079	-0.056
Yurakaré	-0.198	-0.204	-0.191

Table 4: The effect of sequence length and element/interval position on element/interval duration for each human language, computed from the expanded model that includes both length and position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
	Length			Position
Language	Effect	2.5%	97.5%	Effect	2.5%	97.5%
Anal	-0.105	-0.114	-0.096	0.1	0.091	0.109
Arapaho	0.03	0.02	0.039	0.099	0.09	0.109
Asimjeeg Datooga	-0.063	-0.074	-0.053	0.186	0.177	0.196
Baïnounk Gubëeher	-0.102	-0.11	-0.093	0.096	0.088	0.104
Beja	-0.066	-0.073	-0.059	0.065	0.058	0.072
Bora	-0.127	-0.138	-0.116	-0.131	-0.14	-0.123
Cabécar	-0.11	-0.12	-0.1	0.021	0.012	0.031
Cashinahua	-0.1	-0.108	-0.091	-0.019	-0.028	-0.01
Daakie	-0.131	-0.141	-0.122	0.164	0.155	0.173
Dalabon	-0.079	-0.091	-0.066	0.165	0.153	0.177
Dolgan	-0.13	-0.139	-0.121	0.043	0.034	0.052
English (Southern England)	-0.053	-0.066	-0.04	0.05	0.038	0.062
Evenki	-0.101	-0.109	-0.092	0.042	0.033	0.05
Fanbyak	-0.091	-0.101	-0.081	0.161	0.151	0.171
French (Swiss)	-0.05	-0.063	-0.037	0.16	0.151	0.169
Goemai	-0.124	-0.138	-0.11	0.063	0.052	0.074
Gorwaa	-0.125	-0.136	-0.114	0.009	0	0.018
Hoocąk	-0.099	-0.109	-0.09	0.141	0.132	0.151
Jahai	-0.062	-0.073	-0.05	0.142	0.131	0.153
Jejuan	-0.093	-0.104	-0.083	0.038	0.028	0.049
Kakabe	-0.123	-0.135	-0.111	0.103	0.091	0.115
Kamas	-0.096	-0.111	-0.082	0.003	-0.012	0.017
Komnzo	-0.081	-0.091	-0.071	0.026	0.018	0.035
Light Warlpiri	-0.144	-0.154	-0.133	0.078	0.067	0.088
Lower Sorbian	-0.078	-0.089	-0.067	0.046	0.037	0.056
Mojeño Trinitario	-0.147	-0.156	-0.137	-0.074	-0.083	-0.065
Movima	-0.014	-0.023	-0.006	-0.053	-0.062	-0.045
Nafsan (South Efate)	-0.072	-0.082	-0.062	0.094	0.085	0.103
Nisvai	-0.064	-0.074	-0.054	0.151	0.144	0.159
Nllng	-0.098	-0.111	-0.086	0.155	0.142	0.167
Northern Alta	-0.069	-0.079	-0.059	0.091	0.081	0.101
Northern Kurdish (Kurmanji)	-0.022	-0.033	-0.011	0.033	0.023	0.042
Pnar	-0.021	-0.035	-0.007	0.087	0.076	0.099
Resígaro	-0.079	-0.089	-0.069	0.012	0.003	0.022
Ruuli	-0.039	-0.048	-0.03	0.069	0.061	0.078
Sadu	-0.124	-0.136	-0.112	0.2	0.189	0.212
Sanzhi Dargwa	-0.135	-0.148	-0.122	0.012	0	0.023
Savosavo	-0.052	-0.062	-0.042	-0.125	-0.134	-0.117
Sümi	-0.09	-0.101	-0.08	0.109	0.098	0.12
Svan	-0.066	-0.074	-0.057	-0.026	-0.035	-0.017
Tabaq (Karko)	-0.213	-0.223	-0.203	-0.071	-0.08	-0.061
Tabasaran	-0.138	-0.151	-0.125	-0.025	-0.037	-0.012
Teop	-0.17	-0.181	-0.159	0.072	0.063	0.081
Texistepec Popoluca	-0.07	-0.081	-0.059	0.02	0.011	0.03
Urum	-0.126	-0.134	-0.118	-0.007	-0.015	0.001
Vera'a	-0.152	-0.163	-0.141	0.169	0.16	0.177
Warlpiri	-0.147	-0.157	-0.137	-0.026	-0.035	-0.017
Yali (Apahapsili)	-0.192	-0.203	-0.18	0.14	0.13	0.15
Yongning Na	-0.128	-0.142	-0.114	0.098	0.085	0.112
Yucatec Maya	-0.067	-0.079	-0.056	-0.075	-0.084	-0.066
Yurakaré	-0.198	-0.204	-0.191	-0.035	-0.04	-0.029

4 Words in Sentences

The 95% confidence intervals for the effect of sequence length (top) and position (bottom) on element/interval duration for the 16 whale species and 51 human languages. The human language data are comprised of words within sentences.

Figure 2: The 95% confidence intervals for the effect of sequence length (top) and position (bottom) on element/interval duration for the 16 whale species and 51 human languages. The human language data are comprised of words within sentences.

5 Production Constraint Model

James et al. (1) recently found that Menzerath’s law can be detected in pseudorandom sequences of birdsong syllables that are forced to match the durations of real songs. James et al. (1) interpret their model as approximating simple motor constraints, while stronger effects in the real data would indicate additional mechanisms (e.g., communicative efficiency through behavioral plasticity). I originally planned to compare the strength of Menzerath’s law in the real data with simulated data from the model of James et al. (1), as I recently did for house finch song (2), but analyses of language data suggest that it is far too conservative of a null model. 0 of the 51 of languages in the DoReCo dataset exhibit Menzerath’s law to a greater extent than simulated data. Even though many whale species exhibit Menzerath’s law to a greater extent than simulated data from the null model of James et al. (1) (75%; 12 out of 16 species), I do not want to over-interpret this result given the pattern in the human data. Upon further reflection I think that the fundamental assumption of James et al. (1), that sequence durations are governed by motor constraints alone, is unlikely to apply to many species with more complex communication systems. In humpback whales and sperm whales, for example, there appears to be significant inter-individual variation in song and coda length depending on social context (3,4). More details about this analysis are below.

The production constraint model of James et al. (1) works as follows. For each iteration of the model, a pseudorandom sequence was produced for each real song in the dataset. Syllables were randomly sampled (with replacement) from the population until the duration of the random sequence exceeded the duration of the real song. If the difference between the duration of the random sequence and the real song was <50% of the duration of the final syllable, then the final syllable was kept in the sequence. Otherwise, it was removed. Each iteration of the model produces a set of random sequences with approximately the same distribution of durations as the real data.

For each species, I generated 100 simulated datasets from the (1) random sequence model and the (2) production constraint model. Then, I fit Menzerath’s law separately to each of the 100 simulated datasets and pooled the model estimates for \(a\) and \(b\) using Rubin’s rule as implemented in the mice package in R (5). The results can be seen in Figure 3.

Most importantly, the estimated effects from the production constraint model tend to be more negative than those from the real human language data, suggesting that this null model is far too conservative to be informative about “language-like” efficiency.

The point estimates from the real data alongside 95% confidence intervals from 10 simulated datasets from the production constraint model, for the effect of sequence length on element/interval duration for the 16 whale species and 51 human languages. The human language data are comprised of phonemes within words.

Figure 3: The point estimates from the real data alongside 95% confidence intervals from 10 simulated datasets from the production constraint model, for the effect of sequence length on element/interval duration for the 16 whale species and 51 human languages. The human language data are comprised of phonemes within words.

6 Median Interpolation

The point estimates from the original datasets (red) compared to median-interpolated datasets (blue). Interpolating sequences with the median duration of each element category appears to systematically shift model estimates towards zero (in over 90% of cases).

Figure 4: The point estimates from the original datasets (red) compared to median-interpolated datasets (blue). Interpolating sequences with the median duration of each element category appears to systematically shift model estimates towards zero (in over 90% of cases).

7 Patterned Burst Pulses

Martin et al. (6) noticed that Heaviside’s dolphins produce temporally-patterned burst pulses with much more rhythmic variation in some social situations. I analyzed the 27 patterned burst pulses recorded by Martin et al. (6) and found that they adhere to Menzerath’s law—there is a negative relationship between sequence length and element duration (estimate = -0.186, 95% CI: [-0.308, -0.063]).

References

1.

James LS, Mori C, Wada K, Sakata JT. Phylogeny and mechanisms of shared hierarchical patterns in birdsong. Current Biology [Internet]. 2021 Jul [cited 2023 Apr 12];31(13):2796–2808.e9. Available from: https://linkinghub.elsevier.com/retrieve/pii/S0960982221005285

2.

Youngblood M. Language-like efficiency and structure in house finch song. Proceedings of the Royal Society B. 2024;291(2020):20240250.

3.

Mercado E. Intra-individual variation in the songs of humpback whales suggests they are sonically searching for conspecifics. Learn Behav [Internet]. 2022 Dec [cited 2024 Apr 18];50(4):456–81. Available from: https://link.springer.com/10.3758/s13420-021-00495-0

4.

Hersh TA, Gero S, Rendell L, Cantor M, Weilgart L, Amano M, et al. Evidence from sperm whale clans of symbolic marking in non-human cultures. Proc Natl Acad Sci USA [Internet]. 2022 Sep 13 [cited 2024 Apr 15];119(37):e2201692119. Available from: https://pnas.org/doi/full/10.1073/pnas.2201692119

5.

Buuren SV, Groothuis-Oudshoorn K. mice: multivariate imputation by chained equations in R. J Stat Soft [Internet]. 2011 [cited 2024 Mar 8];45(3). Available from: http://www.jstatsoft.org/v45/i03/

6.

Martin MJ, Elwen SH, Kassanjee R, Gridley T. To buzz or burst-pulse? The functional role of Heaviside’s dolphin, Cephalorhynchus heavisidii, rapidly pulsed signals. Animal Behaviour [Internet]. 2019 Apr [cited 2024 Mar 8];150:273–84. Available from: https://linkinghub.elsevier.com/retrieve/pii/S0003347219300089

Supplementary information

Language-like efficiency in whale communication

Mason Youngblood